### 3.1 Geometry

General relativity in its canonical formulation [6] describes the geometry of space-time in terms of fields
on spatial slices. Geometry on such a spatial slice is encoded in the spatial metric , which presents
the configuration variables. Canonical momenta are given in terms of extrinsic curvature which is the
derivative of the spatial metric under changing the spatial slice. Those fields are not arbitrary since they are
obtained from a solution of Einstein’s equations by choosing a time coordinate defining the spatial slices,
and space-time geometry is generally covariant. In the canonical formalism this is expressed by the presence
of constraints on the fields, the diffeomorphism constraint and the Hamiltonian constraint. The
diffeomorphism constraint generates deformations of a spatial slice or coordinate changes, and when it is
satisfied spatial geometry does not depend on which coordinates we choose on space. General
covariance of space-time geometry also for the time coordinate is then completed by imposing the
Hamiltonian constraint. This constraint, furthermore, is important for the dynamics of the theory:
Since there is no absolute time, there is no Hamiltonian generating evolution, but only the
Hamiltonian constraint. When it is satisfied, it encodes correlations between the physical fields of
gravity and matter such that evolution in this framework is relational. The reproduction of
a space-time metric in a coordinate dependent way then requires to choose a gauge and to
compute the transformation in gauge parameters (including the coordinates) generated by the
constraints.
It is often useful to describe spatial geometry not by the spatial metric but by a triad which
defines three vector fields which are orthogonal to each other and normalized in each point. This
yields all information about spatial geometry, and indeed the inverse metric is obtained from
the triad by where we sum over the index counting the triad vector fields.
There are differences, however, between metric and triad formulations. First, the set of triad
vectors can be rotated without changing the metric, which implies an additional gauge freedom
with group SO(3) acting on the index . Invariance of the theory under those rotations is
then guaranteed by a Gauss constraint in addition to the diffeomorphism and Hamiltonian
constraints.

The second difference will turn out to be more important later on: We can not only rotate the triad
vectors but also reflect them, i.e., change the orientation of the triad given by . This does not
change the metric either, and so could be included in the gauge group as O(3). However, reflections are not
connected to the unit element of O(3) and thus are not generated by a constraint. It then has to be seen
whether or not the theory allows to impose invariance under reflections, i.e., if its solutions are reflection
symmetric. This is not usually an issue in the classical theory since positive and negative orientations on the
space of triads are separated by degenerate configurations where the determinant of the metric vanishes.
Points on the boundary are usually singularities where the classical evolution breaks down
such that we will never connect between both sides. However, since there are expectations that
quantum gravity may resolve classical singularities, which indeed are confirmed in loop quantum
cosmology, we will have to keep this issue in mind and not restrict to only one orientation from the
outset.